Newton–Raphson iteration

Numerical methods

A-level Maths (9MA0)

ORIGINAL

Form the Newton–Raphson recurrence from a function and derivative, then iterate without premature rounding.

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Question

Use the Newton–Raphson method with

x_0=2

to solve

x^3-2x-5=0

. Calculate

x_1

x_2

and

x_3

, and give the root to 3 decimal places.

Every step worked, with the reasoning.

1
$f(x)=x^3-2x-5$ and $f'(x)=3x^2-2$
Identify the function and differentiate it for Newton–Raphson.
2
$x_{n+1}=x_n-\dfrac{x_n^3-2x_n-5}{3x_n^2-2}$
Substitute $f$ and $f'$ into $x_{n+1}=x_n-f(x_n)/f'(x_n)$ .
3
$x_1=2-\dfrac{2^3-2(2)-5}{3(2^2)-2}=2.1$
Use $x_0=2$ in the recurrence.
4
$x_2=2.1-\dfrac{2.1^3-2(2.1)-5}{3(2.1^2)-2}=2.094568\ldots$
Feed the unrounded value of $x_1$ back into the recurrence.
5
$x_3=2.094551\ldots$
Repeat once more; the first three decimal places have stabilised.

Answer: $x\approx2.095$ (3 d.p.)